A unified construction yielding precisely Hilbert and James sequences spaces

📝 Abstract

Following James’ approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \in \Bbb{R}^d$ with $ e_1 \ne 0 $. The construction immediately implies that J(1) coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J $. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2 $, if $e_1+e_2+...+e_d\ne 0$ (ii) or $J(e)$ is isomorphic to $J $. Such a dichotomy also holds for every separable Orlicz sequence space $l_M $.

💡 Analysis

Following James’ approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \in \Bbb{R}^d$ with $ e_1 \ne 0 $. The construction immediately implies that J(1) coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J $. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2 $, if $e_1+e_2+...+e_d\ne 0$ (ii) or $J(e)$ is isomorphic to $J $. Such a dichotomy also holds for every separable Orlicz sequence space $l_M $.

📄 Content

arXiv:0804.3131v1 [math.GN] 19 Apr 2008 A UNIFIED CONSTRUCTION YIELDING PRECISELY HILBERT AND JAMES SEQUENCES SPACES Duˇsan Repovˇs and Pavel V. Semenov Abstract. Following James’ approach, we shall define the Banach space J(e) for each vector e = (e1, e2, …, ed) ∈Rd with e1 ̸= 0. The construction immediately implies that J(1) coincides with the Hilbert space i2 and that J(1; −1) coincides with the celebrated quasireflexive James space J. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either J(e) is isomorphic to l2 ,if e1 + e2 + … + ed ̸= 0 (ii) or J(e) is isomorphic to J. Such a dichotomy also holds for every separable Orlicz sequence space lM. 0. Introduction In infinite-dimensional analysis and topology – in Banach space theory, two se- quences spaces – the Hilbert space l2 and the James space J – are certainly presented as a two principally opposite objects. In fact, the Hilbert space is the ”simplest” Banach space with a maximally nice analytical, geometrical and topological prop- erties. On the contrary, the properties of the James space are so unusual and unexpected that J is often called a ”space of counterexamples” (see [3,5]). Let us list some of the James space properties: (a) J has the Schauder basis, but admits no isomorphic embedding into a space with unconditional Schauder basis [1,3,4]; (b) J and its second conjugate J∗∗are separable, but dim(J∗∗/χ(J)) = 1, where χ : J →J∗∗is the canonical embedding (see [1]); (c) in spite of (b), the spaces J and J∗∗are isometric with respect to an equivalent norm (see [2]); (d) J and J ⊕J are non-isomorphic and moreover,J and B ⊕B are non-isomorphic for an arbitrary weakly complete B (see [3, 4]); (e) on J there exists a C1-function with bounded support, but there are no C2-functions with bounded support (see [7]); (f) there exists an infinite-dimensional manifold modelled on J which cannot be homeomorphically embedded into J (see [4,7]); and (g) the group GL(J) of all invertible continuous operators of J onto itself is homotopically non-trivial with respect to the topology generated by operator’s norm (see [8]), but it is contractible in pointwise convergency operator topology (see [10] and the book [3] for more references). In this paper we shall define the Banach space J(e) for each vector e = (e1, e2, …, ed) ∈Rd with e1 ̸= 0. The construction immediately implies that J(1) = l2 and 1991 Mathematics Subject Classification. Primary: 54C60, 54C65, 41A65; Secondary: 54C55, 54C20. Key words and phrases. Hilbert space;Banach space; James sequence space; Invertible contin- uous operator. Typeset by AMS-TEX 1 2 DUˇSAN REPOVˇS AND PAVEL V. SEMENOV J(1; −1) = J. Surprisingly, there are only these two possibilities, up to an isomor- phism. It appears that J(e) is isomorphic to l2, if e1 +e2+…+ed ̸= 0 (see Theorem 5) and J(e) is isomorphic to J otherwise (see Theorem 6). Such a dichotomy holds not only for the space l2 (which is clearly defined by using the numerical function M(t) = t2, t ≥0), but also for an arbitrary Orlicz sequence space lM defined by an arbitrary Orlicz function M : [0; +∞) →[0; +∞) with the so-called ∆2-condition. Then there are also exactly two possibilities for J(e): either J(e) is isomorphic to lM, or J(e) is isomorphic to the James-Orlicz space JM (see [9]). For simplicity, we restrict ourselves below for M(t) = t2, t ≥0.

1. Preliminaries Let d be a natural number and e = (e1, e2, …, ed) ∈Rd a d-vector with e1 ̸= 0. Having in our formulae many brackets we shall choose the special notation a ∗b for the usual scalar product of two elements a ∈Rd and b ∈Rd. A d-subset ω of N is defined by setting ω = {n(1) < n(2) < … < n(d) < n(d + 1) < … < n(kd −1) < n(kd)} ⊂N for some natural k and then the subsets ω(1) = {n(1) < n(2) < … < n(d)}, ω(2) = {n(d+1) < n(d+2) < … < n(2d)}, ……, ω(k) = {n((k−1)d+1) < … < n(kd)} are called the d-components of the set d-set ω. For each d -set ω and each infinite sequence of reals x = (x(m))m∈N ∈RN we denote x(ω) = (x(m))m∈ω and x(ω; i) = (x(m))m∈ω(i). Definition 1. For each d ∈N, e ∈Rd, x ∈RN and d-set ω = {n(1) < … < n(kd)} the (e, ω)-variation of x is defined by the equality (e, ω) = v u u t k X i=1 (e ∗x(ω; i))2 = p (e1x(n(1)) + … + edx(n(d)))2 + … + (e1x(n((k −1)d + 1)) + … + edx(n(kd)))2 Definition 2. For each d ∈N, e ∈Rd, x ∈RN the e-variation of x is defined by the equality ||x||e = sup{e(x, ω) : ω are d-subset of N}. Definition 3. The set of all infinite sequences of reals tending to zero with finite e-variation is denoted by J(e). We omit the routine verification of the following proposition. Proposition 4. (J(e); || · ||e) is a Banach space for each e = (e1, e2, …, ed) ∈Rd with e1 ̸= 0. □ Note that a restriction e1 ̸= 0 is purely technical. It avoids the case e = 0 ∈Rd and guarantees that∥(1; 0; 0; …)∥e > 0. Below we fix such a hypotesis. A UNIFIED CONSTRUCTION 3 Theorem 5. If e1 + e2 + … + ed ̸= 0, then J(e) and l2 are i

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